HIGH-LEVEL DESIGN OF INTEGRATED MICROSYSTEMS - ARITHMETIC
PERSPECTIVE
Zeljko Zilic
Boris Karajica
McGill University, Dept. ECE
Montréal, Québec, Canada
STMicroelectronics
Boston, Massachusets, USA
ABSTRACT
Integration of physical sensing and actuating interfaces into
embedded systems presents a new set of high-level design and synthesis challenges. An important aspect of such
integration is the explicit use of numerical values at the
interfaces, so we propose the system design method focusing
on the efficient and correct arithmetic-centric synthesis.
Integrated precision and range analysis and optimization are
detailed, together with the iterative arithmetic processing.
Index Terms— cyber-physical systems, microsystems,
arithmetic optimization, static arithmetic analysis
I. INTRODUCTION
Cyber-physical systems are characterized by tighter integration of embedded computer systems and the physical
sensing and actuating devices. A significant part of the
appeal of cyber-physical systems is in the integration of
heterogeneous subsystems, be it electronics, MEMS, microfluidics, and at the same time the functionality provided
in increasingly sophisticated software. Such integrated microsystems will need to be designed and optimized following
the design techniques that are suited for complete systems.
For this reason, high-level design tools and methodologies
will need to be developed that account for the new issues in
interactions among the subsystems.
For instance, consider a robot including a MEMS-based
accelerometer with a given precision tolerance, temperature
dependence and frequency characteristics. There is a need
to optimize interfaces and the embedded control, and to
tailor them to compensate for the imperfections in physical
sensors. In this scenario, we ought to optimize the overall
system design, rather than apply a series of isolated steps that
either insert the pessimism (and hence are inefficient) or are
not adequate for the whole range of inputs and states of the
systems (and hence are incorrect under some conditions).
In trying to undertake the high-level design and optimization of heterogeneous microsystems, we are necessarily
bound to deal with the continuous-valued variables in the
system, such as the analog outputs of the sensors. The
issue we are facing here is that the physical processes
are continuous in nature, both in time and the state-space,
while the computing part is discrete-valued and executed in
discrete time steps. Therefore, the required integration has to
address salient features of interfaces between the discrete and
continuous domains in providing optimization, verification,
guaranteeing fidelity in dealing with physical processes as
well as the overall quality measures in integrated system
design. [1].
I-A. Arithmetic-centric Microsystem Synthesis
The methodologies that we explore deal with the efficient
and correct handling of the numerical quantities used in
integrated microsystems. The ”control plane” is assumed to
be largely equivalent to that of the systems on chip. The
explicit measures are undertaken to provide the end-to-end
functionality guarantees for:
• Sufficient dynamic range of all quantities of interest,
such that when digitized we employ the bit-widths
that are both sufficient to avoid overflows and also
efficiently use the area, energy and time for discretized
computation. Calibration will also deal with range.
• Adequate precision of the results of computation across
the integrated microsystem, given by some acceptable
measures of the distance to the specified output, or, in
other words, the fidelity of the system operation.
• Suitable response in the frequency domain, taking into
account the behavior of the mechanical, chemical,
electronic and the discrete-time control subsystems.
Frequency response includes the requirements for the
range and precision above.
• Interaction and fusion of multiple sources of data, including noise filtering. As the sampled data is bound to
be noisy and might appear inconsistent, solid techniques
need to be applied at the system level.
II. INTEGRATED MICROSYSTEM SYNTHESIS
An increasing range of system designs, from embedded
systems to consumer electronics to FPGA-based prototypes
now require tight integration of physical sensors and actuators. The design methodologies for such systems will
need to be established, with the clear aim of achieving an
overarching integrated system-level design. With the tight
integration into physical processes, the specification of such
systems is necessarily given in terms of the system dynamics
in continuous space and time, as well as in frequency
domain. We illustrate the issues in such a system design
using the example.
Example 1: Avionics tilt-compensated compass. Consider a system including a 3D accelerometer, magnetometer
and gyroscope that is specified to compute the tilt angles
α, β and γ in three dimensions, with the latter being the
exact position of magnetic north, known as heading. The
heading calculation should operate under normal and freefall conditions, in which case the interrupt service routine
(ISR) is required to calculate the heading direction, using
an alternative algorithm that does not rely on the accelerometer. The system should be self-calibrated, and tilt angles
calculated with the precision of 2 degrees. The acceleration
and the magnetic signal noise in all three dimensions should
be filtered above fa and fH cutoff frequencies, respectively.
The three readings of acceleration ax , ay and az are used
to obtain the Euler angles, relative to the three coordinate
planes as:


a
x

(1)
α = arctan  q
(a2y + a2z )
!
ay
β = arctan p
(2)
(a2x + a2z )
Since the obtained accelerometer readings do not suffice
for obtaining the third angle, which is heading, as that
information is invariant relative to the gravity force. For
this angle, the magnetometer reading is used to obtain
compensations in X and Y directions and the heading angle
γ.
X 0 = X cos(α) − Y sin(β) sin(α) + Z cos(β) sin(α)
Y0 =
γ=
Y cos(β) + Z sin(α)
X0
90 − arctan( 0 ), when Y 0 ≥ 0
Y
We consider the integrated design of the whole system,
and note that the MEMS sensors can already be easily
added to the CMOS substrate performing arbitrary computing and communication functions. Companies such as ST
Microelectronics routinely perform 3D integration of such
systems including multiple discrete MEMS sensors, hence
the system-level synthesis is a clear possibility.
Numerous discrete MEMS sensors are in existence that
sense all quantities in Example 1. For example, the acceleration sensor measurements in the immobile state corresponds
to the three projections of the gravity force to the coordinate
axes. The magnetic module similarly measures the three projections of the magnetic field H. Figure 1 shows the details of
the integrated geomagnetic module by ST Microelectronics,
with 3 accelerometers and 3 magnetometers as well as the
interface circuitry [2].
Fig. 1. Example of integrated geomagnetic sensor module
in 3 dimensions: LSM303DLM by ST Microelectronics [2]
II-A. Correct and Precise Arithmetic Computation
We focus on the integration of numerical quantities in
the synthesis of integrated microsystems. In Example 1, the
main component of the system is that of calculating the tilt
angles within given precision constraints, which entails the
selection of the calculation algorithm, optimization of the
word-lengths and avoidance of the incorrect cases, be it due
to the range (overflow), corner cases or the noise.
To obtain the required precision, the system design needs
to account for all the causes of imprecision across the
mechanical parts (ereadout , tempcomp ), analog electronics
(econv ) and the finite word-length digital processing parts.
Errorα = ereadout + econv + ∆t ∗ tempcomp + earithmetic
The synthesis task then entails finding suitable word-lengths
and possibly approximation algorithms for the last part
earithmetic , such that the overall error bound is not exceeded:
Errorα ≤ .
(3)
While doing this optimization, we also need to ensure the
arithmetic correctness, including the avoidance of arithmetic
overflows, conditions when denominator approaches zero
and similar, which exceed the range of possible arithmetic
representations.
II-B. Noise, Self-calibration and Data Fusion
The noise exhibits significant presence in the sensor
data, and handling noisy data is critical to all applications
of microsystems, ranging from the usable user interfaces
in consumer electronics employing motion sensors, to the
emerging cyber-physical systems. Considering Example 1,
with specified cutoff frequencies for the accelerometer and
magnetometer measurements, one can easily appreciate that
any mechanical vibrations introduce unwanted noise in all
three dimensions, hence the requirement to eliminate all
the sampled accelerometer signal above given frequency fa .
Similarly, magnetometers are constantly impacted by numerous artificially produced magnetic fields present everywhere,
and the data readouts would appear wrong without proper
filtering.
Often, sensor modules such as the one from Figure 1
already come equipped with digital filters. Further, the
frequency characteristics of the sensor system might itself
play a significant role, or a set of filters could be available
as IP cores for synthesis. For all such scenarios, in order to
integrate and filter the data, the system design would require
the fusion synthesis, to cascade the newly synthesized modules with the existing transfer characteristics, to obtain the
required frequency response, Figure 2. The problem could
involve a solution in hardware, software or a combination,
calling for the solution space exploration.
Sensor HW SW |H| Filter/IP Sensor … … Filter/IP ? f Fig. 2. Fusion synthesis: fitting frequency-domain response
The utility of multiple sensor systems is impossible without the noise-free integration of all sensed data, referred to
as data fusion, whereas solutions such as Kalman filters are
just one possibility, and in their implementation, the same
word-length issues (precision, range) remain, but now are
additionally applied to the iterative and non-linear computations. System validation, including the runtime correctness
now involves the effects due to the physical device effects in
sensors, and the self-calibration, temperature compensation
etc. are all must in interfacing the sensors.
III. ARITHMETIC-CENTERED MICROSYSTEM
SYNTHESIS
The arithmetic issues in high-level synthesis are longstanding [3], [4], [5], with the noticeable renewal in interest
due to the proliferation of FPGAs as ASIC replacements,
which force the use of word-length limited fixed-point representations.
Suitable results in arithmetic-driven high-level synthesis
and verification include the methods for word optimizations
under precision constraints [6] that employs Arithmetic
Transform (AT) [7], [8]. For range analysis and optimization,
there is the satisfiability modulo theory (SMT) approach[9].
III-A. Arithmetic Verification and Optimization with AT
AT is defined as a polynomial over pseudo-Boolean
functions, f : B n 7→ w, where the output w is a wordlevel quantity, over which we can perform addition and
subtraction [10].
Table I. AT Encoding for Common Arithmetic Functions
Word
Integer
n−1
P
Unsign.
Fixed-Point
n−1
P
xi 2i
i=0
Sign-ext.
(1 − 2xn−1 )
n−1
P
xi 2i
i=0
2 Compl.
n−2
P
xi 2i−m
i=1
xi 2i − xn−1 2n−1
(1 − 2x0 )
n−1
P
−x0 2m−n −
1 X
1
X
xi 2i−m
i=1
i=0
f=
xi 2i−m
i=1
n−2
P
1
X
···
i0 =0 i1 =0
i
n−1
.
ci0 i1 ...in−1 xi00 xi11 . . . xn−1
(4)
in−1 =0
Hence, AT expresses a function using the set of linearly
in−1
independent functions defined as: xi00 xi11 . . . xn−1
, where
(
xj , if ij = 1
i
, j = 0, . . . , n − 1.
xjj =
1, if ij = 0
We say that the arithmetic spectrum is a set of coefficients
ci0 i1 ...in−1 , each of which multiplies an orthogonal basis
in−1
function xi00 xi11 . . . xn−1
. In the case of AT, the transform
can be simply treated as a polynomial.
An important propertiy of AT is that it allows the outputs
to be grouped into the word-level quantities, which allows us
to directly reason about the arithmetic, including determining
statically the precision and range values. Then, the encoding of word-level quantities determines the exact way the
calculation is done, i.e., the meaning of the operations ”+”
and ”−”, to provide the ”numerical” value . The common
encodings are summarized in Table I. The same encoding
can be easily expanded to describe more complex functions
as shown next.
III-B. Relation to Real-valued Specifications
Arithmetic circuits commonly implement real-valued
specifications of functions that are in general multivariate.
For a number of real-valued specifications, including polynomials and Taylor series, there is an efficient construction
of AT.
For function f over interval I around point X0 , the infinite
Taylor series converging over I are:
f (X) =
∞
X
f (i) (X0 )
i=1
i!
∗ (X − X0 )i .
(5)
In practical implementations, the series become truncated
to the first n + 1 terms, in which case the remainder is
provably bounded by an (n + 1st ) order derivative:
Rn (X) =
f (n+1) (ξ)
∗ (X − X0 )n+1
(n + 1)!
(6)
where ξ is a point in the given interval I. The finite Taylor
expansion can be converted efficiently to an AT [11]. For
an instance of m-bit fixed-point presentation the arithmetic
rewrite replaces all instances of the real-valued X with the
encoding function over a m-tuple of bits. For instance, with
unsigned encoding, X0 = 0 and f0 = f (X0 ), the AT for the
(n + 1)-term Taylor polynomial is:

n
!
m−1
m−1
(n)
X
X
0
f
f0 + f0
xj 2j + · · · + 0 ∗ 
xj 2 j 
n!
i=0
j=0
The single most challenging issue in the conversion to
AT is that of the intermediate polynomial swell during the
transformation. The techniques for the efficient rewriting are
well elaborated in, for instance, symbolic computing area.
Having the efficient construction of AT from real-valued
specification, we proceed to the precision analysis, which
will allow us to efficiently verify whether a fixed-point
implementation achieves the required precision.
III-C. AT and Arithmetic Precision and Range
Implementations of real-valued functions are bound to be
imprecise. First, there are the approximations applied to the
real-valued specification, and the imprecision is based on a
finite-word implementation.
The imprecision error is a distance between the specification and implementation, most commonly measured as
the maximal absolute difference between the two, i.e., the
uniform norm L∞ . The error is commonly expressed in the
terms of the units in the last place (ULP), i.e., relative to
the length of the fixed-point implementation.
The precision analysis cannot be dealt with at the bit-level,
and the explicit representation of output word-level values
is needed. Consider an example of computation where all
output bits are incorrect, while the imprecision can be made
arbitrarily small by extending the wordlength. For example,
if the exact n-bit fractional result is 1.00 . . . 0,, while the
approximation is 0.11 . . . 1., then all bits are incorrect, but
the error can be made arbitrarily small by increasing n.
Since AT represents the function output at a word level,
it can be used for reasoning directly about the imprecision.
For a given reference implementation fref , the imprecision
due to the n-bit wordlength is then directly expressed as
AT (fref ) − AT (fn ).
Hence, the imprecision to the finite wordlength n is expressed in terms of maximum absolute distance in L∞ as
||AT (fref ) − AT (fn )||∞ = ||AT (fref − fn )||∞ .
(7)
Then, it suffices to search for a maximum absolute value
that the AT of a difference between the reference and
the implementation functions take via an branch-and-bound
search [12]. Furthermore, the other distance functions, like
the sum of squares could be equally applied as well.
To verify an implementation consisting of arithmetic values passed by multiple sensors, we construct the AT and
Problem 1 P RECISION V ERIFICATION
Require: fref , fimpl , ε
Ensure: T RUE iff ||AT (fref ) − AT (fimpl )||∞ < ε
evaluate whether the imprecision from Eqn. 7 is acceptable,
i.e., smaller than a given bound ε. The problem is given as:
This formulation is also practical if a reference fref by
itself is imprecise within a bound, say δ. For instance,
the transcendental functions can only have the approximate
reference implementations, such as with Taylor series. Then,
we can apply the triangle inequality
||AT (fabs ) − AT (fimpl )||∞ ≤ ||AT (fabs ) − AT (fref )||∞
+||AT (fref ) − AT (fimpl )||∞
among the precise fabs , the reference and the implementation, to guarantee that |ε + δ| is an acceptable imprecision.
Hence, AT facilitates a completely static verification
scheme precision and range constraints. Towards that goal,
a branch-and-bound search in [12] finds whether the worst
case imprecision of a fixed-point implementation is smaller
than the allowed imprecision. The exact search for the
maximal imprecision is reduced to a search for a maximal
value that an AT polynomial takes, as per Eqn. 7.
For range analysis, it suffices to find the minimal and
maximal value of the AT, which is directly achieved by
the same search algorithm applied to the implementation
polynomial.
III-D. Component Matching and Fusion Synthesis
Another task in synthesis is that of finding the most
suitable library components for imprecise arithmetic, including the transductor synthesis, Figure II-B. The component
matching has surfaced as a problem in conjunction with
IP library reuse. The work such as [13] has shown the
need for multivariate polynomials real-valued polynomials
for the matching of real-valued functions. With AT, the transition to multivariate real-valued polynomial specifications
is straightforward. Further, the search for a suitable library
component from a given library lib is reduced to the problem
P RECISION V ERIFICATION, applied to all the elements of
lib.
III-E. Optimization of Fixed-point Arithmetic
We show next how to optimize the fixed-point implementations of integrated microsystems. Starting from a polynomial or Taylor series description, the algorithm automatically
selects the multiple approximation and imprecision parameters by a scheme that is essentially branch-and-bound, with
a number of optimizations specific to this case.
While this approach could be applied to a variety of realvalued representations, we present the main ideas applied
to the case of finite Taylor expansion. The optimization
problem is defined as that of finding the parameters such
as the number of Taylor terms, n, and the wordlength m of
the input representation, such that the cost is minimal.
Problem 2 TAYLOR P RECISION O PTIMIZATION
Require: fref , ε
Ensure: ||AT (fref ) − AT (fn,m )||∞ < ε
Goal: m, n s.t. min terms(AT (fn,m ))
Consider now the number of AT terms of an expanded nth
order Taylor polynomial over m-bit fixed point numbers as a
cost function in the optimization procedure. By enumeration,
it can be shown that the number of AT terms is then equal
to
n X
m
cost = terms(AT (fn,m )) =
i
i=1
In this case, the cost function exhibits the same monotonicity
in the two variables. By using this cost function, we will
effectively be searching for a minimal size AT polynomial
implementing a function within a given precision.
Search guidance - sensitivity Similar to the previous instance of branch-and-bound, providing a search heuristic
is a significant way to speed up the algorithm. In this
case, we can apply the domain-specific knowledge about the
sensitivity of a function relative to the change in either n or
m.
Sensitivity of function f with respect to variable x is
df
. Here, variables n
defined as a derivative of a function dx
and m are discrete, and need to be mapped to continuous
variables that they present.
In the case of m, the wordlength, an increase of one bit
results in the added shift of half an ulp: ∆X = 2−(m+1) .
Then, the impact of the wordlength change to the function
is
df
df −(m+1)
∆f =
∆X =
2
.
dX
dX
Furthermore, by recalling that the first Taylor term is a
derivative at the expansion point X0 , the sensitivity is
calculated as an m+1-fold shift of the first coefficient. Since
this is true only around X0 , a more accurate sensitivity is
obtained by differentiating the function. AT can be used here
as well, to easily differentiate the given function. By finding
the maximal value over the interval, as in the precision
search, the worst case influence of the variable change is
obtained.
In the case of sensitivity to n, we apply the error bound
from Eqn. 6. When a new n is to be selected in the search,
the difference between the two remainder functions, bounding approximation error, can be easily computed. Further,
when going, say, from n to n + 1 terms, the function
difference is
∆f = T aylorn+1 (f ) − T aylorn (f ) + Rn (f ) − Rn+1 (f )
The algorithm takes into account in an unified way all
the approximation and imprecision sources for real-valued
specifications such as Taylor series, where only the parameters n and m are to be chosen. The pseudo-code is given
in Algorithm 3. It starts with the number of terms given
by Eqn. 6 and then explores m and n according to the
sensitivity.
Algorithm 3 O PTIMIZE (f, ε)
1: lb = M IN TAYLORT ERMS
2: CurrentBest = lb; Current = ub;
3: M IN A REA (AT (f ), Current)
4: {
5: if U nexploredP osibilities then
6:
direction = F IND M OST S ENSITIVE VAR
7:
if (direction ==m) then
8:
Current = M IN A REA (AT (fm ), Current)
9:
else
10:
Current = M IN A REA (AT (fn ), Current)
11:
end if
12: else
13:
Current = AT ;
14:
CurrentBest = |min(Current, CurrentBest)|
15:
return CurrentBest
16: end if
17: }
This algorithm allows us to perform design space exploration of microsystems, which are arithmetic-intensive by
finding the precise implementation within given approximation schemes (Taylor, polynomial, lookup table, etc.).
IV. EXPERIMENTAL RESULTS
We have undertaken the design space exploration and
arithmetic optimization of the system outlined in Example 1,
as carried out in the project [14]. For each of its subsystems,
multiple designs were completed and the results were compared towards finding the most suitable solution targeting
the iNEMO board [2]. Because of the presence of the ARM
Cortex M3 processor running at 72MHz, we showcase for
comparison here three obvious candidates suited for the
software implementations, leaving out the CORDIC and
other schemes suited more for hardware.
First, the implementations of the arctan function and the
whole block for computing the three angles in Eqn. 1 were
explored for ARM Cortex software implementation. The
times it takes to execute the function for the best and worst
case, as reported by Keil ARM development system, are
reported in Table II. The code developed by the existing C
library is included for comparison. The table lookup method
requires 400 bytes of on-chip FLASH memory and it is
precise within 1.15o . The approximation property, Eqn. 3 for
achieving the overall specified precision was verified using
AT static analysis from [6] in negligible time.
Table II. Exploration of Sufficiently Precise arctan function
Input Case
Pol. Appr.
LUT
C library
Best
84.4 µs
14.6 µs
305.3 µs
Worst
103 µs
22.8 µs
311.6 µs
Table III. Filters/transductors Meeting Spec
Max Rate
1st Order/50
3rd Order/100
1st Recur/50
Sample
Data
Output
2.4kHz
1.96 kHz
2.4 kHz
62.3 Hz
124.6 Hz
58 Hz
31.1 Hz
31.1 Hz
31.1 Hz
Next, the design exploration of the fusion/filter was conducted, taking into account that the sensors already come
equipped with a lowpass (and highpass, which is in this
case turned off) filter. In this case, the lowest cutoff rate
for the lowpass filter is 37Hz, so the frequency-domain
fusion/transductor needs to be inserted to merge and filter
further to fa =12 Hz. Among the range of solutions explored
within the project, we show here non-recursive and recursive
first order mean filter and third order mean filter, sampled
at 50 or 100 Hz. Its performance in the ARM Cortex code
for the target output rate at 31 Hz is given in Table III.
The overall performance is obtained for the selection
of the LUT-based arctan calculation and the third order
filter, as reported in Table IV. Here, the specification of the
sampling rates were given (except that the free-fall interrupts
were processed asynchronously), the actual processing time
includes the data transfer through all hardware interfaces
(such as I 2 C for accelerometer and magnetometer) and the
code performing the readouts, filtering, fusion and overall
processing.
V. CONCLUSIONS AND FUTURE WORK
We demonstrate that the AT allows static analysis and
optimization for realizing full-scale synthesis of multi-sensor
systems with the emphasis on the numerical values passed
between sensors and the rest of the system, and their correct
and efficient processing. We have identified the key steps for
the static analysis for precision, range and iterative arithmetic
computation based on AT. We will consider fault-tolerant
design [15] of multisensor systems.
Table IV. System Performance
Operation
Time [µs]
Rate [Hz]
Accel. Readout
2761
100
Magnet. Readout
1250
100
Filter accel.
35
25
Filter magn.
75
25
Free-fall ISR
280
async
Free-fall filter
452
25
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